Understanding the Slope of a Line

The slope of a line is a measure of its steepness. It tells us how much the vertical position (y-coordinate) changes for every unit change in the horizontal position (x-coordinate). The slope is often denoted by the letter 'm'.

Formula for Calculating Slope

To find the slope of a straight line passing through two distinct points, say \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), we use the following formula:

\(\displaystyle m = \frac{\text{Change in y}}{\text{Change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\)

Applying the Slope Formula to the Given Points

We are asked to find the slope of the line joining the points (3, -4) and (5, 2).

Let's identify our points and their coordinates:

Now, we substitute these coordinates into the slope formula:

\(\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}\)

\(\displaystyle m = \frac{2 - (-4)}{5 - 3}\)

First, calculate the difference in the y-coordinates (\(y_2 - y_1\)):

\(2 - (-4) = 2 + 4 = 6\)

Next, calculate the difference in the x-coordinates (\(x_2 - x_1\)):

\(5 - 3 = 2\)

Now, divide the change in y by the change in x:

\(\displaystyle m = \frac{6}{2}\)

\(\displaystyle m = 3\)

Resulting Slope Calculation

The slope of the line joining the points (3, -4) and (5, 2) is 3.

Revision Table: Slope Calculation Key Points

Additional Information on Linear Equations

Understanding the slope is fundamental in coordinate geometry and linear equations. Here are some related concepts:

• Types of Slopes: A line can have a positive slope (rises from left to right), a negative slope (falls from left to right), a zero slope (horizontal line), or an undefined slope (vertical line).
• Equation of a Line: The slope-intercept form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept (the point where the line crosses the y-axis).
• Parallel Lines: Two distinct non-vertical lines are parallel if and only if they have the same slope.
• Perpendicular Lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. A vertical line and a horizontal line are also perpendicular.